Problem 17 - Intermediate Algebra

Question

Solve each inequality. $$ 3 x^{2}+13 x-10 \leq 0 $$

Step-by-Step Solution

Answer

The solution is \([-5, \frac{2}{3}]\).

Step 1: Identify the Parabola

Recognize that the inequality involves a quadratic expression, which represents a parabola. The inequality is of the form \(3x^2 + 13x - 10 \leq 0\). This means we are looking for the values of \(x\) where the quadratic is less than or equal to zero.

Step 2: Solve the Quadratic Equation

First, solve the quadratic equation \(3x^2 + 13x - 10 = 0\) to find the roots of the parabola. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 13\), and \(c = -10\).

Step 3: Quadratic Formula Calculation

Calculate the discriminant: \(b^2 - 4ac = 13^2 - 4 \cdot 3 \cdot (-10) = 169 + 120 = 289\). Since the discriminant is positive, we have two real roots. Calculate the roots: \(x = \frac{-13 \pm \sqrt{289}}{6}\).

Step 4: Determine the Roots

The square root of 289 is 17, so the roots are \(x_1 = \frac{-13 + 17}{6} = \frac{4}{6} = \frac{2}{3}\) and \(x_2 = \frac{-13 - 17}{6} = \frac{-30}{6} = -5\).

Step 5: Parabola Analysis

The roots \(x = \frac{2}{3}\) and \(x = -5\) divide the xy-plane into intervals. The parabola opens upwards (as \(a = 3 > 0\)), so the quadratic is negative (or zero) between the roots.

Step 6: Determine Solution Interval

Since the parabola opens upwards, the solution of \(3x^2 + 13x - 10 \leq 0\) is the interval where the parabola is below or on the x-axis. Thus, the solution is the closed interval \([-5, \frac{2}{3}]\).

Key Concepts

Quadratic Formula

Understanding the quadratic formula is a crucial step in solving quadratic inequalities. It is a special equation solution technique used to find the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is defined as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This powerful tool allows us to solve any quadratic equation as long as we can identify the coefficients \(a\), \(b\), and \(c\). In the given inequality \(3x^2 + 13x - 10 \leq 0\), the coefficients are \(a = 3\), \(b = 13\), and \(c = -10\). By substituting these values into the quadratic formula, we can find the roots, which are the values of \(x\) that satisfy the equation. Calculating the discriminant, \(b^2 - 4ac\), is vital as it tells us the nature of the roots. A positive discriminant indicates two real roots, which is the case here with a value of 289.

Parabola Analysis

Once we find the roots using the quadratic formula, we can analyze the parabola represented by the quadratic inequality. A parabola is a symmetrical curve that opens upwards or downwards. In this scenario, the parabola represented by \(3x^2 + 13x - 10\) opens upwards since the leading coefficient \(a = 3\) is positive. Knowing this helps us determine where the parabola lies in relation to the x-axis. The parabola crosses the x-axis at the roots \(x_1 = \frac{2}{3}\) and \(x_2 = -5\). Between these points, we need to identify where the function is less than or equal to zero.

Solution Interval

The solution interval is the range of \(x\) values that make the inequality \(3x^2 + 13x - 10 \leq 0\) true. After analyzing the parabola, we know the regions where the parabola is below or touches the x-axis are of interest. Since the parabola opens upwards, the section between the two roots \([-5, \frac{2}{3}]\) is where the quadratic function is less than or equal to zero. This interval represents the solution to the inequality. Important points about the interval:

• The inequality includes \(\leq \), which means the roots themselves are part of the solution.
• The interval is closed, denoted by brackets \([-5, \frac{2}{3}]\), indicating \(x = -5\) and \(x = \frac{2}{3}\) satisfy the inequality.

Roots of Equation

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. For the equation \(3x^2 + 13x - 10 = 0\), we calculated two distinct roots: \(x_1 = \frac{2}{3}\) and \(x_2 = -5\). These roots can also be thought of as the x-intercepts of the parabola associated with the quadratic equation.Key details about roots:

• The roots split the real number line into regions where the quadratic expression is either positive or negative.
• In the context of an inequality, these roots help us decide over which intervals the quadratic expression satisfies the inequality \(\leq 0\).

Understanding the roots helps clearly establish how the parabola interacts with the x-axis and supports finding the correct solution interval.